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They do not stack because the damage roll of Flames of Phlegethos is separate from any spell's damage roll

The first bullet of Flames of Phlegethos states:

When you roll fire damage for a spell you cast, you can reroll any roll of 1 on the fire damage dice and use the new roll.

And thus it can only apply to damage rolls of spells you cast. The other bullet however, states:

Whenever you cast a spell that deals fire damage ... until the end of your next turn  ... any creature within 5 feet of you that hits you with a melee attack takes 1d4 fire damage.

The damage from this is not you casting a spell, it simply happens when you are hit by a melee attack. The 1d4 additional damage is not part of the spell's damage because, though it does rely on the spell to exist, it is not part of the spell's description. The extra 1d4 fire damage are the feat's own damage roll.

Because these dice are not part of the spell's damage roll these parts of the feat do not stack with each other.

They do not stack because the damage roll of Flames of Phlegethos is separate from any spell's damage roll

The first bullet of Flames of Phlegethos states:

When you roll fire damage for a spell you cast, you can reroll any roll of 1 on the fire damage dice and use the new roll.

And thus it can only apply to damage rolls of spells you cast. The other bullet however, states:

Whenever you cast a spell that deals fire damage you can cause flames to wreathe around you until the end of your next turn.  .. any creature within 5 feet of you that hits you with a melee attack takes 1d4 fire damage.

The damage from this is not you casting a spell, it simply happens when you are hit by a melee attack. Even though the spell "causes flames to wreathe around you" , these flames are not considered part of the spell as they are not in its description. This let's us establish that the damage the flames cause is not part of the spell's damage roll. Thus the parts of this feat cannot be used together because the additional 1d4 will always be its own damage roll, separate from any spell's.

They do not stack because the damage roll of Flames of Phlegethos is separate from any spell's damage roll

The first bullet of Flames of Phlegethos states:

When you roll fire damage for a spell you cast, you can reroll any roll of 1 on the fire damage dice and use the new roll.

And thus it can only apply to damage rolls of spells you cast. The other bullet however, states:

Whenever you cast a spell that deals fire damage ... until the end of your next turn ... any creature within 5 feet of you that hits you with a melee attack takes 1d4 fire damage.

The damage from this is not you casting a spell, it simply happens when you are hit by a melee attack. The 1d4 additional damage is not part of the spell's damage because, though it does rely on the spell to exist, it is not part of the spell's description. The extra 1d4 fire damage are the feat's own damage roll.

Because these dice are not part of the spell's damage roll these parts of the feat do not stack with each other.

This is the case even with the spell fire shield which states:

In addition, whenever a creature within 5 feet of you hits you with a melee attack, the shield erupts with flame. The attacker takes 2d8 fire damage

Even though both Flames of Phlegethos and fire shield have damage rolls with the same trigger they would still be separate damage rolls. This is because Flames of Phlegethos's damage is not in addition to or on top of some other source damage like Sneak Attack is:

Once per turn, you can deal an extra 1d6 damage to one creature you hit with an attack

Because Flames of Phlegethos is always its own damage roll, it will never be part of a spell's damage roll. Thus even if flame shield were active the feat's properties still would not benefit each other.

They do not stack because the damage roll of Flames of Phlegethos is separate from any spell's damage roll

The first bullet of Flames of Phlegethos states:

When you roll fire damage for a spell you cast, you can reroll any roll of 1 on the fire damage dice and use the new roll.

And thus it can only apply to damage rolls of spells you cast. The other bullet however, states:

Whenever you cast a spell that deals fire damage ... until the end of your next turn ... any creature within 5 feet of you that hits you with a melee attack takes 1d4 fire damage.

The damage from this is not you casting a spell, it simply happens when you are hit by a melee attack. The 1d4 additional damage is not part of the spell's damage because, though it does rely on the spell to exist, it is not part of the spell's description. The extra 1d4 fire damage are the feat's own damage roll.

Because these dice are not part of the spell's damage roll these parts of the feat do not stack with each other.

They do not stack because the damage roll of Flames of Phlegethos is separate from any spell's damage roll

The first bullet of Flames of Phlegethos states:

When you roll fire damage for a spell you cast, you can reroll any roll of 1 on the fire damage dice and use the new roll.

And thus it can only apply to damage rolls of spells you cast. The other bullet however, states:

Whenever you cast a spell that deals fire damage ... until the end of your next turn ... any creature within 5 feet of you that hits you with a melee attack takes 1d4 fire damage.

The damage from this is not you casting a spell, it simply happens when you are hit by a melee attack. The 1d4 additional damage is not part of the spell's damage because, though it does rely on the spell to exist, it is not part of the spell's description. The extra 1d4 fire damage are the feat's own damage roll.

Because these dice are not part of the spell's damage roll these parts of the feat do not stack with each other.

There is the spell fire shield which states:

In addition, whenever a creature within 5 feet of you hits you with a melee attack, the shield erupts with flame. The attacker takes 2d8 fire damage

Even though both Flames of Phlegethos and fire shield would have their damage rolls happening from the same trigger they would still be separate damage rolls. This is because Flames of Phlegethos's damage is not in addition to or on top of some other source damage like Sneak Attack is:

Once per turn, you can deal an extra 1d6 damage to one creature you hit with an attack

The damage from Flames of Phlegethos is not extra or added to another damage roll and so it is its own damage roll, separate from any spell. Thus even if flame shield were active the feat's properties still would not benefit each other.

They do not stack because the damage roll of Flames of Phlegethos is separate from any spell's damage roll

The first bullet of Flames of Phlegethos states:

When you roll fire damage for a spell you cast, you can reroll any roll of 1 on the fire damage dice and use the new roll.

And thus it can only apply to damage rolls of spells you cast. The other bullet however, states:

Whenever you cast a spell that deals fire damage ... until the end of your next turn ... any creature within 5 feet of you that hits you with a melee attack takes 1d4 fire damage.

The damage from this is not you casting a spell, it simply happens when you are hit by a melee attack. The 1d4 additional damage is not part of the spell's damage because, though it does rely on the spell to exist, it is not part of the spell's description. The extra 1d4 fire damage are the feat's own damage roll.

Because these dice are not part of the spell's damage roll these parts of the feat do not stack with each other.

This is the case even with the spell fire shield which states:

In addition, whenever a creature within 5 feet of you hits you with a melee attack, the shield erupts with flame. The attacker takes 2d8 fire damage

Even though both Flames of Phlegethos and fire shield have damage rolls with the same trigger they would still be separate damage rolls. This is because Flames of Phlegethos's damage is not in addition to or on top of some other source damage like Sneak Attack is:

Once per turn, you can deal an extra 1d6 damage to one creature you hit with an attack

Because Flames of Phlegethos is always its own damage roll, it will never be part of a spell's damage roll. Thus even if flame shield were active the feat's properties still would not benefit each other.